Simplify and expand the following expression: $ \dfrac{4y}{y - 8}+\dfrac{y + 8}{y + 8} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(y - 8)(y + 8)$ Multiply the first term by $\dfrac{y + 8}{y + 8}$ $ \begin{align*} \dfrac{4y}{y - 8} \times \dfrac{y + 8}{y + 8} & = \dfrac{(4y)(y + 8)}{(y - 8)(y + 8)} \\ & = \dfrac{4y^2 + 32y}{(y - 8)(y + 8)}\end{align*} $ Multiply the second term by $\dfrac{y - 8}{y - 8}$ $ \begin{align*} \dfrac{y + 8}{y + 8} \times \dfrac{y - 8}{y - 8} & = \dfrac{(y + 8)(y - 8)}{(y + 8)(y - 8)} \\ & = \dfrac{y^2 - 64}{(y + 8)(y - 8)}\end{align*} $ Now we have: $ = \dfrac{4y^2 + 32y}{(y - 8)(y + 8)} + \dfrac{y^2 - 64}{(y + 8)(y - 8)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{4y^2 + 32y + y^2 - 64}{(y - 8)(y + 8)} $ $ = \dfrac{5y^2 + 32y - 64}{(y - 8)(y + 8)}$ Expand the denominator: $ = \dfrac{5y^2 + 32y - 64}{y^2 - 64}$